A new method of determining the critical state in superconductors

Abstract

A new numerical method for solving the critical state, based on the force–displacement curve of the flux lines, is described. The equation can be expressed in terms of the vector potential and can be solved by commercial finite element programs. It gives the critical state directly as a first approximation but a flux flow resistivity can easily be added. It avoids some of the numerical problems that occur if an E–J curve of the form is used with large values of n. It is particularly advantageous for problems involving trapped flux where a power law leads to decay at a rate dependent on the power assumed and the very low electric fields mean that the E–J curve is probably exponential. The flux flow resistivity can be included and since the relevant power law is 1/n rather than n the solutions are more stable. Multiphysics packages, such as FlexPDE, which allow simultaneous equations to be solved, can find the temperature distribution by adding the heat flow equation.

Flux avalanches in Nb superconducting shifted strip arrays

“…3(f)). The electromagnetic responses of SSAs are calculated by using the Campbell method with assuming the Bean model [39]. Figure 5 shows the flux distribution in SSA300-50 with a x = (a) 9 µm and (b) 13 µm under applied field of H = 0.1J c d s , where J c is the critical current density.…”

Theoretical simulation studies of pulsed field magnetisation of (RE)BCO bulk superconductors

“…An approach developed by A. M. Campbell in 2007 [7], based on the force-displacement curve of magnetic flux in HTS [8,9], has been adopted in the present model. This approach solves directly the critical state, in which J = J c wherever magnetic flux penetrates the sample.…”

“…Here A p and J p are the magnetic vector potential and the corresponding current density of the initial state, respectively, A r is regarded as a constant dependent on the material when a sufficiently 'large uniform' field is applied to the superconductor [7], and ρ v is the flux flow resistivity of the (RE)BCO bulk material. It is worth pointing out that the normal state resistivity of the material is assumed for ρ v and that the model is not based on an E-J power law relationship since the electric field is extremely high.…”

Potential and limits of numerical modelling for supporting the development of HTS devices

“…Another particularly original approach is the force-displacement model developed by Campbell [17], which starts from a physical argument and reduces the critical Figures adapted from [8].…”

“…As an example, Campbell's force-displacement method [17] is naturally written in terms of the magnetic vector potential A, so not using the A − V formulation in this case would be cumbersome.…”